The descendants of three U.S. presidents and the nation`s best-known civil rights activist are seeking political office this year, and so are the kids of some recent presidential hopefuls. There are more famous names on the ballot this year than in any election since 1970, when Edward M. Kennedy easily won re-election to the U.S. Senate, and Adlai Stevenson III and Robert A. Taft Jr. were also elected senators.

In the 8th Congressional District of Massachusetts, where House Speaker Tip O`Neill is retiring this year, Joseph P. Kennedy III, son of the late Sen. Robert F. Kennedy and nephew of President John F. Kennedy, is running against a field that includes James Roosevelt Jr., the grandson of Franklin D. Roosevelt. On different terrain, the older and more experienced Roosevelt might have had a chance. But in the heavily Democratic district that includes Cambridge and much of Boston, the Kennedys are as popular as the Red Sox and Celtics. Kennedy, 33, has made no secret of his national political ambitions. In Georgia, the scion of another celebrated family, Martin Luther King III, has declared his candidacy for the state Senate seat being vacated by Julian Bond, now running for Congress. Without holding public office, the son and namesake of the late civil rights leader has become well known as a spokesman for progressive issues and his family`s rich legacy. Like Joe Kennedy, King will be recognized as among the Democratic Party`s rising stars if he is elected this year.

Former Gov. James Rhodes of Ohio, who is attempting a political comeback against Democratic Gov. Richard Celeste, strengthened the GOP ticket by naming Robert A. Taft III, a product of the state`s best-known political family, as his running-mate for lieutenant governor. Taft, the great-grandson of President William Howard Taft, and the son and grandson of two U.S. senators, is viewed as among the more promising rookies of the 1986 political season. But his GOP relatives have lost as many elections as they have won in recent years, and it will be a comeback for the Tafts as well as Rhodes if they can upset Celeste.

Former Sen. Adlai E. Stevenson III went from slight favorite to underdog in his campaign for governor of Illinois after two of his running-mates were upset by followers of political eccentric Lyndon LaRouche in the March Democratic primary. Stevenson, whose father served as governor from 1949 to 1953 and was twice the Democratic presidential nominee in the 1950s, has been coveting the governor`s mansion for almost 20 years. Before his setback in the primary, Stevenson confided that, like his father, he too was interested in running for the presidency.

Stevenson`s name recognition and nostalgia for his father`s memory helped the former senator squeeze out Atty. Gen. Neil Hartigan for the gubernatorial nomination. But Stevenson, who trails Republican Gov. Jim Thompson in his own poll, is finding that it takes more than name identity to win a general election.

Lt. Gov. William W. Scranton III of Pennsylvania is another namesake interested in following in his father`s footsteps. Scranton, 39, is facing a tough Democratic opponent, former State Auditor Bob Casey, who turned down an invitation from Scranton`s father to become the GOP nominee for governor more than a decade ago. Scranton`s father, William II, was a popular and highly effective governor and unsuccessful candidate for the 1964 GOP presidential nomination. The younger Scranton is described by associates as more ambitious than his father. But to the Republican Party`s far right, Scranton is about as well-liked as the late Nelson A. Rockefeller.

Atty. Gen. Hubert H. Humphrey III of Minnesota is a strong favorite to win re-election and is viewed by many political observers as the best Democratic hope for recapturing the U.S. Senate seat long held by his late father, Hubert H. Humphrey.

In Indiana, Evan Bayh, son of former Sen. Birch Bayh, is the Democratic nominee for secretary of state this year, and is considered the brightest prospect on the state ticket.

And in Chicago, one of the biggest questions in political circles is whether Cook County State`s Atty. Richard M. Daley will declare his candidacy later this year against Mayor Harold Washington and former Mayor Jane Byrne, setting up a rematch of their 1983 Democratic mayoral primary. Daley is plainly interested in the job his father had for so long. But maybe he`s waiting to determine if it`s the right season for famous political names.